April 5, 2017 Subject:
Manifolds for beginners
This is an introduction to manifolds approached from a mostly a multivariable calculus / analysis direction. Thus it is most suitable for students who have just had a one-semester course in one-variable analysis. Spivak's text is a little bit more sophisticated, proving fewer theorems explicitly than Munkres and asking students to make the appropriate generalizations to proofs they've already seen. However, Spivak is also slightly sloppy at times with respect to rigor, and he has numerous typos and errors to boot. Students who want to most seamless transition to the study of manifolds should strongly consider Munkres (despite the wordiness and penchant to spoonfeed results), while students who want an in depth treatment should consider Loomis and Sternberg. Spivak's book looks cool, but no one should use it as their exclusive textbook to learn about forms and manifolds.